Finite-dimensional sensor orbits and optimal nonlinear filtering

The filtering problem of a system with linear dynamics and non-Gaussian a priori distribution is investigated. A closed-form exact solution to the problem is presented along with an approximation scheme. The approximation is made in the construction of a mathematical model. It reduces optimal estimation to a combination of linear estimations. The asymptotic behavior of the filter is examined. The limiting distributions of the conditional mean and the conditional-error covariance exist as the time interval of observation becomes infinite. In the autonomous case, the estimate for the Wiener problem satisfies a linear stochastic differential equation. A large class of nonlinear problems with more nonlinear features than the one discussed above can be reduced to it through the idea of finite-dimensional sensor orbits. The general idea and a number of examples are discussed briefly.